Structure and models of real-world graphs and networks

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Structure and models of real-world graphs and
networks

• Jure Leskovec
• Machine Learning Department
• Carnegie Mellon University
• jure_at_cs.cmu.edu
• http//www.cs.cmu.edu/jure/

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Networks (graphs)
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Examples of networks
(b)
(c)
(a)
(d)
(e)

• Internet (a)
• citation network (b)
• World Wide Web (c)

• sexual network (d)
• food web (e)

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Networks of the real-world (1)

• Information networks
• World Wide Web hyperlinks
• Citation networks
• Blog networks
• Social networks people interactios
• Organizational networks
• Communication networks
• Collaboration networks
• Sexual networks
• Collaboration networks
• Technological networks
• Power grid
• Airline, road, river networks
• Telephone networks
• Internet
• Autonomous systems

Florence families
Karate club network
Collaboration network
Friendship network
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Networks of the real-world (2)

• Biological networks
• metabolic networks
• food web
• neural networks
• gene regulatory networks
• Language networks
• Semantic networks
• Software networks

Semantic network
Yeast protein interactions
Language network
XFree86 network
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Types of networks

• Directed/undirected
• Multi graphs (multiple edges between nodes)
• Hyper graphs (edges connecting multiple nodes)
• Bipartite graphs (e.g., papers to authors)
• Weighted networks
• Different type nodes and edges
• Evolving networks
• Nodes and edges only added
• Nodes, edges added and removed

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Traditional approach

• Sociologists were first to study networks
• Study of patterns of connections between people
  to understand functioning of the society
• People are nodes, interactions are edges
• Questionares are used to collect link data (hard
  to obtain, inaccurate, subjective)
• Typical questions Centrality and connectivity
• Limited to small graphs (10 nodes) and
  properties of individual nodes and edges

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New approach (1)

• Large networks (e.g., web, internet, on-line
  social networks) with millions of nodes
• Many traditional questions not useful anymore
• Traditional What happens if a node U is removed?
  
• Now What percentage of nodes needs to be removed
  to affect network connectivity?
• Focus moves from a single node to study of
  statistical properties of the network as a whole
• Can not draw (plot) the network and examine it

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New approach (2)

• How the network looks like even if I cant look
  at it?
• Need statistical methods and tools to quantify
  large networks
• 3 parts/goals
• Statistical properties of large networks
• Models that help understand these properties
• Predict behavior of networked systems based on
  measured structural properties and local rules
  governing individual nodes

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Statistical properties of networks

• Features that are common to networks of different
  types
• Properties of static networks
• Small-world effect
• Transitivity or clustering
• Degree distributions (scale free networks)
• Network resilience
• Community structure
• Subgraphs or motifs
• Temporal properties
• Densification
• Shrinking diameter

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Small-world effect (1)

• Six degrees of separation (Milgram 60s)
• Random people in Nebraska were asked to send
  letters to stockbrokes in Boston
• Letters can only be passed to first-name
  acquantices
• Only 25 letters reached the goal
• But they reached it in about 6 steps
• Measuring path lengths
• Diameter (longest shortest path) max dij
• Effective diameter distance at which 90 of all
  connected pairs of nodes can be reached
• Mean geodesic (shortest) distance l

or
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Small-world effect (2)

• Distribution of shortest path lengths
• Microsoft Messenger network
• 180 million people
• 1.3 billion edges
• Edge if two people exchanged at least one message
  in one month period

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Small-world effect (3)

• Fact
• If number of vertices within distance r grows
  exponentially with r, then mean shortest path
  length l increases as log n
• Implications
• Information (viruses) spread quickly
• Erdos numbers are small
• Peer to peer networks (for navigation purposes)
• Shortest paths exists
• Humans are able to find the paths
• People only know their friends
• People do not have the global knowledge of the
  network
• This suggests something special about the
  structure of the network
• On a random graph short paths exists but no one
  would be able to find them

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Transitivity or Clustering

• friend of a friend is a friend
• If a connects to b, and b to c, then with
  high probability a connects to c.
• Clustering coefficient C
• C 3number of triangles / number of connected
  triples
• Alternative definition
• Ci triangles connected to vertex i / number
  triples centered on vertex i
• Clustering coefficient

Ci1, 1, 1/6, 0, 0
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Transitivity or Clustering (2)

• Clustering coefficient scales as
• It is considerably higher than in a random graph
• It is speculated that in real networks
• CO(1) as n?8
• In Erdos-Renyi random graph CO(n-1)

Synonyms network
World Wide Web
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Degree distributions (1)

• Let pk denote a fraction of nodes with degree k
• We can plot a histogram of pk vs. k
• In a Erdos-Renyi random graph degree distribution
  follows Poisson distribution
• Degrees in real networks are heavily skewed to
  the right
• Distribution has a long tail of values that are
  far above the mean
• Heavy (long) tail
• Amazon sales
• word length distribution,

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Detour how long is the long tail?
This is not directly related to graphs, but it
nicely explains the long tail effect. It shows
that there is big market for niche products.
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Degree distributions (2)

• Many real world networks contain hubs highly
  connected nodes
• We can easily distinguish between exponential and
  power-law tail by plotting on log-lin and log-log
  axis
• In scale-free networks maximum degree scales as
  n1/(a-1)

lin-lin
log-lin
pk
k
k
log-log
pk
k
Degree distribution in a blog network
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Poisson vs. Scale-free network
Poisson network
Scale-free (power-law) network
(Erdos-Renyi random graph)
Degree distribution is Power-law
Function is scale free if f(ax) b f(x)
Degree distribution is Poisson
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Degree distribution number of people a person
talks to on a Microsoft Messenger
Count
Highest degree
X
Node degree
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Network resilience (1)

• We observe how the connectivity (length of the
  paths) of the network changes as the vertices get
  removed
• Vertices can be removed
• Uniformly at random
• In order of decreasing degree
• It is important for epidemiology
• Removal of vertices corresponds to vaccination

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Network resilience (2)

• Real-world networks are resilient to random
  attacks
• One has to remove all web-pages of degree gt 5 to
  disconnect the web
• But this is a very small percentage of web pages
• Random network has better resilience to targeted
  attacks

Random network
Internet (Autonomous systems)
Preferential removal
Mean path length
Random removal
Fraction of removed nodes
Fraction of removed nodes
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Community structure

• Most social networks show community structure
• groups have higher density of edges within than
  accross groups
• People naturally divide into groups based on
  interests, age, occupation,
• How to find communities
• Spectral clustering (embedding into a low-dim
  space)
• Hierarchical clustering based on connection
  strength
• Combinatorial algorithms
• Block models
• Diffusion methods

Friendship network of children in a school
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MSN Messenger
Distribution of Connected components in MSN
Messenger network
Growth of largest component over time in a
citation network
Count

• Graphs have a giant component
• Distribution of connected components follows a
  power law

Largest component
X
Size (number of nodes)
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Network motifs (1)

• What are the building blocks (motifs) of
  networks?
• Do motifs have specific roles in networks?
• Network motifs detection process
• Count how many times each subgraph appears
• Compute statistical significance for each
  subgraph probability of appearing in random as
  much as in real network

3 node motifs
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Network motifs (2)

• Biological networks
• Feed-forward loop
• Bi-fan motif
• Web graph
• Feedback with two mutual diads
• Mutual diad
• Fully connected triad

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Network motifs (3)
Transcription networks
Signal transduction networks
WWW and friendship networks
Word adjacency networks
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Networks over time Densification

• A very basic question What is the relation
  between the number of nodes and the number of
  edges in a network?
• Networks are becoming denser over time
• The number of edges grows faster than the number
  of nodes average degree is increasing
• a densification exponent 1 a 2
• a1 linear growth constant out-degree (assumed
  in the literature so far)
• a2 quadratic growth clique

Internet
E(t)
a1.2
N(t)
Citations
E(t)
a1.7
N(t)
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Densification degree distribution
Degree exponent over time

• How does densification affect degree
  distribution?
• Given densification exponent a, the degree
  exponent is
• (a) For ?const over time, we obtain
  densification only for 1lt?lt2, then ?a/2
• (b) For ?lt2 degree distribution has to evolve
  according to
• Power-law yb x?, for ?lt2 Ey 8

pkk?
(a)
?(t)
a1.1
(b)
?(t)
a1.6
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Shrinking diameters
Internet

• Intuition says that distances between the nodes
  slowly grow as the network grows (like log n)
• But as the network grows the distances between
  nodes slowly decrease

Citations
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Models of network generation and evolution
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Recap (1)

• Last time we saw
• Large networks (web, on-line social networks) are
  here
• Many traditional questions not useful anymore
• We can not plot the network so we need
  statistical methods and tools to quantify large
  networks
• 3 parts/goals
• Statistical properties of large networks
• Models that help understand these properties
• Predict behavior of networked systems based on
  measured structural properties and local rules
  governing individual nodes

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Recap (2)

• We also so features that are common to networks
  of various types
• Properties of static networks
• Small-world effect
• Transitivity or clustering
• Degree distributions (scale free networks)
• Network resilience
• Community structure
• Subgraphs or motifs
• Temporal properties
• Densification
• Shrinking diameter

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Outline for today

• We will see the network generative models for
  modeling networks features
• Erdos-Renyi random graph
• Exponential random graphs (p) model
• Small world model
• Preferential attachment
• Community guided attachment
• Forest fire model
• Fitting models to real data
• How to generate a synthetic realistic looking
  network?

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(Erodos-Renyi) Random graphs

• Also known as Poisson random graphs or Bernoulli
  graphs
• Given n vertices connect each pair i.i.d. with
  probability p
• Two variants
• Gn,p graph with m edges appears with probability
  pm(1-p)M-m, where M0.5n(n-1) is the max number
  of edges
• Gn,m graphs with n nodes, m edges
• Very rich mathematical theory many properties
  are exactly solvable

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Properties of random graphs

• Degree distribution is Poisson since the presence
  and absence of edges is independent
• Giant component average degree k2m/n
• k1-e all components are of size log n
• k1e there is 1 component of size n
• All others are of size log n
• They are a tree plus an edge, i.e., cycles
• Diameter log n / log k

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Evolution of a random graph
for non-GCC vertices
k
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Subgraphs in random graphs

• Expected number of subgraphs H(v,e) in Gn,p is

a... of isomorphic graphs
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Random graphs conclusion
Configuration model

• Pros
• Simple and tractable model
• Phase transitions
• Giant component
• Cons
• Degree distribution
• No community structure
• No degree correlations
• Extensions
• Configuration model
• Random graphs with arbitrary degree sequence
• Excess degree Degree of a vertex of the end of
  random edge qk k pk

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Exponential random graphs(p models)

• Comes from social sciences
• Let ei set of measurable properties of a graph
  (number of edges, number of nodes of a given
  degree, number of triangles, )
• Exponential random graph model defines a
  probability distribution over graphs

Examples of ei
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Exponential random graphs

• Includes Erdos-Renyi as a special case
• Assume parameters ßi are specified
• No analytical solutions for the model
• But can use simulation to sample the graphs
• Define local moves on a graph
• Addition/removal of edges
• Movement of edges
• Edge swaps
• Parameter estimation
• maximum likelihood
• Problem
• Cant solve for transitivity (produces cliques)
• Used to analyze small networks

Example of parameter estimates
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Small-world model

• Used for modeling network transitivity
• Many networks assume some kind of geographical
  proximity
• Small-world model
• Start with a low-dimensional regular lattice
• Rewire
• Add/remove edges to create shortcuts to join
  remote parts of the lattice
• For each edge with prob p move the other end to a
  random vertex
• Rewiring allows to interpolate between regular
  lattice and random graph

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Small-world model

• Regular lattice (p0)
• Clustering coefficient C(3k-3)/(4k-2)3/4
• Mean distance L/4k
• Almost random graph (p1)
• Clustering coefficient C2k/L
• Mean distance log L / log k
• No power-law degree distribution

Rewiring probability p
Degree distribution
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Models of evolution

• Models of network evolution
• Preferential attachment
• Edge copying model
• Community Guided Attachment
• Forest Fire model
• Models for realistic network generation
• Kronecker graphs

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Preferential attachment

• Models the growth of the network
• Preferential attachment (Price 1965, Albert
  Barabasi 1999)
• Add a new node, create m out-links
• Probability of linking a node ki is
  proportional to its degree
• Based on Herbert Simons result
• Power-laws arise from Rich get richer
  (cumulative advantage)
• Examples (Price 1965 for modeling citations)
• Citations new citations of a paper are
  proportional to the number it already has

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Preferential attachment

• Leads to power-law degree distributions
• But
• all nodes have equal (constant) out-degree
• one needs a complete knowledge of the network
• There are many generalizations and variants, but
  the preferential selection is the key ingredient
  that leads to power-laws

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Edge copying model

• Copying model
• Add a node and choose k the number of edges to
  add
• With prob ß select k random vertices and link to
  them
• With prob 1-ß edges are copied from a randomly
  chosen node
• Generates power-law degree distributions with
  exponent 1/(1-ß)
• Generates communities
• Related Random-surfer model

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Community guided attachment

• Want to model/explain densification in networks
• Assume community structure
• One expects many within-group friendships and
  fewer cross-group ones

University
Arts
Science
CS
Drama
Music
Math
Self-similar university community structure
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Community guided attachment

• Assuming cross-community linking probability
• The Community Guided Attachment leads to
  Densification Power Law with exponent
• a densification exponent
• b community tree branching factor
• c difficulty constant, 1 c b
• If c 1 easy to cross communities
• Then a2, quadratic growth of edges near
  clique
• If c b hard to cross communities
• Then a1, linear growth of edges constant
  out-degree

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Forest Fire Model

• Want to model graphs that density and have
  shrinking diameters
• Intuition
• How do we meet friends at a party?
• How do we identify references when writing papers?

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Forest Fire Model for directed graphs

• The model has 2 parameters
• p forward burning probability
• r backward burning probability
• The model
• Each turn a new node v arrives
• Uniformly at random chooses an ambassador w
• Flip two geometric coins to determine the number
  in- and out-links of w to follow (burn)
• Fire spreads recursively until it dies
• Node v links to all burned nodes

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Forest Fire Model

• Simulation experiments
• Forest Fire generates graphs that densify and
  have shrinking diameter

E(t)
densification
diameter
1.32
diameter
N(t)
N(t)
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Forest Fire Model

• Forest Fire also generates graphs with
  heavy-tailed degree distribution

in-degree
out-degree
count vs. in-degree
count vs. out-degree
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Forest Fire Parameter Space

• Fix backward probability r and vary forward
  burning probability p
• We observe a sharp transition between sparse and
  clique-like graphs
• Sweet spot is very narrow

Clique-like graph
Increasing diameter
Constant diameter
Sparse graph
Decreasing diameter
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Kronecker graphs

• Want to have a model that can generate a
  realistic graph
• Static Patterns
• Power Law Degree Distribution
• Small Diameter
• Power Law Eigenvalue and Eigenvector Distribution
• Temporal Patterns
• Densification Power Law
• Shrinking/Constant Diameter
• For Kronecker graphs all these properties can
  actually be proven

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Kronecker Product a Graph
Intermediate stage
Adjacency matrix
Adjacency matrix
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Kronecker Product Definition

• The Kronecker product of matrices A and B is
  given by
• We define a Kronecker product of two graphs as a
  Kronecker product of their adjacency matrices

N x M
K x L
NK x ML
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Stochastic Kronecker Graphs

• Create N1?N1 probability matrix P1
• Compute the kth Kronecker power Pk
• For each entry puv of Pk include an edge (u,v)
  with probability puv

Kronecker multiplication
Instance Matrix G2
P1
flip biased coins
P2
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Fitting Kronecker to Real Data

• Given a graph G and Kronecker matrix P1 we can
  calculate probability that P1 generated G
  P(GP1)

P1
G
Pk
P(GP1)
s node labeling
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Fitting Kronecker 2 challenges
P1
G
Pk
P(GP1)

• Invariance to node labeling s (there are N!
  labelings)
• Calculating P(GP1) takes O(N2) (since one needs
  to consider every cell of adjacency matrix)

1
2
3
4

2
1
4
3
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Fitting Kronecker Solutions
s node labeling
P
G

• Node Labeling can use MCMC sampling to average
  over (all) node labelings
• P(GP1) takes O(N2) Real graphs are sparse, so
  calculate P(Gempty) and then add the edges.
  This takes O(E).

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Experiments on real AS graph
Degree distribution
Hop plot
Network value
Adjacency matrix eigen values
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Why fitting generative models?

• Parameters tell us about the structure of a graph
• Extrapolation given a graph today, how will it
  look in a year?
• Sampling can I get a smaller graph with similar
  properties?
• Anonymization instead of releasing real graph
  (e.g., email network), we can release a synthetic
  version of it

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Processes taking place on networks
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Epidemiological processes

• The simplest way to spread a virus over the
  network
• S Susceptible
• I Infected
• R Recovered (removed)
• SIS model 2 parameters
• ß virus birth rate
• d virus death (recovery) rate
• SIR model as one gets cured, he or she can not
  get infected again

SIS model
?it depends on ß and topology
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Epidemic threshold for SIS model

• How infectious the virus needs to be to survive
  in the network?
• First results on power-law networks suggested
  that any virus will prevail
• New result that works for any topology
• For sgt1 virus prevails
• For slt1 virus dies

?1 largest eigen value of graph adjacency matrix
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Navigation in small-world networks
v

• Milgrams experiment showed
• (a) short paths exist in networks
• (b) humans are able to find them
• Assume the following setting
• Nodes of a graph are scattered on a plane
• Given starting node u and we want to reach target
  node v
• A small world navigation algorithm navigates the
  network by always navigating to a neighbor that
  is closest (in Manhattan distance) to target node
  v

u
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Navigation in small-world networks
Network creation

• Start with random lattice
• Each node connects with their 4 immediate
  neighbors
• Long range links are added with probability
  proportional to the distance between the points
  (p(u,v) da)
• Can be show that only for a2 delivery time is
  poly-log in number of nodes n

Deliver time T lt nß
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Navigation in a real-world network

• Take a social network of 500k bloggers where for
  each blogger we know their geographical location
• Pick two nodes at random and geographically
  greedy navigate the network
• Results
• 13 success rate (vs. 18 for Milgram)

Distribution of path lengths
Friendships vs. distance
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Navigation in real-world network

• Geographical distance may not be the right kind
  of distance
• Since population is non-uniform lets use rank
  based friendship distance
• i.e., we measure the distance d(u,v) by the
  number of people living closer to v than u does
• Then

And the proof still works
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• Some references used to prepare this talk
• The Structure and Function of Complex Networks,
  by Mark Newman
• Statistical mechanics of complex networks, by
  Reka Albert and Albert-Laszlo Barabasi
• Graph Mining Laws, Generators and Algorithms, by
  Deepay Chakrabarti and Christos Faloutsos
• An Introduction to Exponential Random Graph (p)
  Models for Social Networks by Garry Robins, Pip
  Pattison, Yuval Kalish and Dean Lusher
• Graph Evolution Densification and Shrinking
  Diameters, by Jure Leskovec, Jon Kleinberg and
  Christos Faloutsos
• Realistic, Mathematically Tractable Graph
  Generation and Evolution, Using Kronecker
  Multiplication, by Jure Leskovec, Deepayan
  Chakrabarti, Jon Kleinberg and Christos Faloutsos
  
• Navigation in a Small World, by Jon Kleinberg
• Geographic routing in social networks, by David
  Liben-Nowell, Jasmine Novak, Ravi Kumar,
  Prabhakar Raghavan, and Andrew Tomkins
• Some plots and slides borrowed from Lada Adamic,
  Mark Newman, Mark Joseph, Albert Barabasi, Jon
  Kleinberg, David Lieben-Nowell, Sergi Valverde
  and Ricard Sole

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Rough random materialthat did not make it into
the presentation
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Bow-tie structure of the web
TENDRILS44M
SCC56 M
OUT44 M
IN44 M
DISC17 M
Broder al. WWW 2000, Dill al. VLDB 2001
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Study of 3 websites

• study over three universities publicly indexable
  Web sites

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Australia
In- and out-degree distributions
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New Zealand
In- and out-degree distributions
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United Kingdom
In- and out-degree distributions
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We assume this node would like to connect to a
centrally located node a node whose distances to
other nodes is minimized.
dij is the Euclidean distance hj is some measure
of the centrality of node j a is a parameter
a function of the final number n of points,
gauging the relative importance of the two
objectives
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Fabrikant et al. define 3 possible measures of
centrality 1. The average number of hops from
other nodes 2. The maximum number of hops from
another node 3. The number of hops from a fixed
center of the tree
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a is the crux of the theorem! Why? Here are
some examples
Fabrikantal
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If a is too low, then the Euclidian distances
become unimportant, and the network resembles a
star
Fabrikantal
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But if a grows at least as fast as vn, where n is
the final number of points, then distance becomes
too important, and minimum spanning trees with
high degree occur, but with exponentially
vanishing probability thus not a power law. if
a is anywhere in between, we have a power law
Through a rather complex and elaborate proof,
Fabrikantal prove this initial assumption will
produce a power law distribution Ill save you
the math!